Statistics: Unlocking the Power of Data Lock5

Section 6.2

Confidence Interval for a Single Proportion

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OutlineConfidence interval for a single proportion

Determining sample size

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SE for

The standard error for is

Problem: when doing inference, we don’t know p!

Solution: substitute , our best guess for p

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Confidence Interval for p

If n is large enough for np ≥ 10 and n(1 – p) ≥ 10, then a confidence

interval for p can be computed by

*

statistic z* SE

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In March 2011, a random sample of 1000 US adults were asked

“Do you favor or oppose ‘sin taxes’ on soda and junk food?”

320 adults responded in favor of sin taxes.

Give a 95% CI for the proportion of all US adults that favor these sin taxes.

Sin Taxes

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Counts are greater than 10 in each category

For a 95% confidence interval, z* = 1.96

Sin Taxes

We are 95% confident that between 29.1% and 34.9% of US adults favor sin taxes on soda and junk food.

*

0.32

0.32 0.015

(0.291, 0.349)

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Sin Taxes

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In a random sample of 500 US citizens, 280 plan to vote in the upcoming election. We plan to construct a 90% confidence interval for the proportion of US citizens who plan to vote. What is the sample proportion?

A. 280B. 500C. 56D. 0.56E. 1.96

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In a random sample of 500 US citizens, 280 plan to vote in the upcoming election. We plan to construct a 90% confidence interval for the proportion of US citizens who plan to vote. What is the value of z*?

A. 0.56B. 1.28C. 1.645D. 1.96E. 2.576

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In a random sample of 500 US citizens, 280 plan to vote in the upcoming election. We plan to construct a 90% confidence interval for the proportion of US citizens who plan to vote. What is the standard error?

A. 0.56B. 500C. 0.00049D. 0.0365E. 0.0222

=0.0222

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In a random sample of 500 US citizens, 280 plan to vote in the upcoming election. We plan to construct a 90% confidence interval for the proportion of US citizens who plan to vote. What is the margin of error?

A. 0.56B. 500C. 0.00049D. 0.0365E. 0.0222

CI = statistic margin of error

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Margin of Error

For a single proportion, what is the margin of error?

a) b) * c) 2 *

CI = statistic margin of error

CI: *

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Margin of Error

You can choose your sample size in advance, depending on your desired margin of error!

Given this formula for margin of error, solve for n.

ME = *

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Margin of Error

n

• Neither p nor is known in advance. To be conservative, use p = 0.5.

• For a 95% confidence interval, z* 2

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Margin of Error

Suppose we want to estimate a proportion with a margin of error of 0.03 with 95% confidence.

How large a sample size do we need?

(a) About 100(b) About 500(c) About 1000(d) About 5000

n

n

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Tongue Curling

What proportion of people can roll their tongue?

Can you roll your tongue? (a) Yes (b) No Visualize and summarize the data. What is your

point estimate? Give and interpret a confidence interval. Tongue rolling has been said to be a dominant trait,

in which case theoretically 75% of all people should be able to roll their tongues. Do our data provide evidence otherwise?